To determine which expressions represent rational numbers, we need to simplify each expression and check if its components are rational numbers. A rational number is any number that can be expressed as the quotient of two integers (i.e., as a fraction), including terminating decimals.
1. [tex]\(\sqrt{100} + \sqrt{100}\)[/tex]:
[tex]\[
\sqrt{100} = 10
\][/tex]
[tex]\[
\sqrt{100} + \sqrt{100} = 10 + 10 = 20
\][/tex]
Since 20 is a rational number, this expression is rational.
2. [tex]\(13.5 + \sqrt{81}\)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
[tex]\[
13.5 + 9 = 22.5
\][/tex]
Since 22.5 is a rational number, this expression is rational.
3. [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]:
[tex]\[
\sqrt{9} = 3
\][/tex]
[tex]\[
\sqrt{729} = 27
\][/tex]
[tex]\[
\sqrt{9} + \sqrt{729} = 3 + 27 = 30
\][/tex]
Since 30 is a rational number, this expression is rational.
4. [tex]\(\sqrt{64} + \sqrt{353}\)[/tex]:
[tex]\[
\sqrt{64} = 8
\][/tex]
[tex]\(\sqrt{353}\)[/tex] is non-integer because 353 is not a perfect square.
[tex]\[
\sqrt{64} + \sqrt{353} = 8 + \sqrt{353}
\][/tex]
Since [tex]\(\sqrt{353}\)[/tex] is an irrational number, this expression is not rational.
5. [tex]\(\frac{1}{3} + \sqrt{216}\)[/tex]:
[tex]\[
\frac{1}{3} = 0.333\ldots
\][/tex]
[tex]\(\sqrt{216}\)[/tex] is non-integer because 216 is not a perfect square.
[tex]\[
\frac{1}{3} + \sqrt{216}
\][/tex]
Since [tex]\(\sqrt{216}\)[/tex] is an irrational number, this expression is not rational.
6. [tex]\(\frac{3}{5} + 2.5\)[/tex]:
[tex]\[
\frac{3}{5} = 0.6
\][/tex]
[tex]\[
2.5 = 2.5 \text{ (rational number since it is a terminating decimal)}
\][/tex]
[tex]\[
\frac{3}{5} + 2.5 = 0.6 + 2.5 = 3.1
\][/tex]
Since 3.1 is a rational number, this expression is rational.
Summarizing the results:
- [tex]\(\sqrt{100} + \sqrt{100}\)[/tex] is rational.
- [tex]\(13.5 + \sqrt{81}\)[/tex] is rational.
- [tex]\(\sqrt{9} + \sqrt{729}\)[/tex] is rational.
- [tex]\(\sqrt{64} + \sqrt{353}\)[/tex] is not rational.
- [tex]\(\frac{1}{3} + \sqrt{216}\)[/tex] is not rational.
- [tex]\(\frac{3}{5} + 2.5\)[/tex] is rational.
Therefore, the expressions that represent rational numbers are:
- [tex]\(\sqrt{100} + \sqrt{100}\)[/tex]
- [tex]\(13.5 + \sqrt{81}\)[/tex]
- [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]
- [tex]\(\frac{3}{5} + 2.5\)[/tex]
